What are the extrema of # f(x)=x/(x-2)# on the interval [-5,5]?

1 Answer
Jan 17, 2016

Answer:

There are no absolute extrema, and the existence of relative extrema depends on your definition of relative extrema.

Explanation:

#f(x) = x/(x-2)# increases without bound as #xrarr2# from the right.
That is: #lim_(xrarr2^+)f(x)=oo#
So, the function has no absolute maximum on #[-5,5]#

#f# decreases without bound as #xrarr2# from the left, so there is no absolute minimum on #[-5,5]#.

Now, #f'(x) = (-2)/(x-2)^2# is always negative, so, taking the domain to be #[-5,2) uu (2,5]# , the function decreases on #[-5,2)# and on #(2,5]#.

This tells us that #f(-5)# is the greatest value of #f# nearby considering only #x# values in the domain. It is a one-sided relative maximum. Not all treatments of calculus allow one sided relative extrema.

Similarly, if your approach allow one-sided relative extrema, then #f(5) is a relative mimimum.

To help visualize, here is a graph. The restricted domain graph is solid and the endpoints are marked.

The natural domain graph extends into the dashed line part of the picture.

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